
where in the third line we again use corollary 2.2.2 in [1]} to conclude that \( ({N}/{k}) \times 1/ U_{k, N}\overset{d}{=}\left( {k}/{N}\right)Y_{N-k, N} \xrightarrow{}1\) ; in the fourth line we write \(\gamma =\gamma ^{\prime }+\varkappa \)  where \(\varkappa >0\)  by assumption, and last line follows by assumptions of the proposition and since \(L\)  is slowly varying.
